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Full text of "Treatise On Analysis Vol-Ii"

15    THE LEBESGUE-NIKODYM THEOREM       197

(b) v = g • ju, where g is locally n-integrable and g(x) > 0 almost every*
where with respect to ju.

If (a) is satisfied, it follows from (13.15.5) that v =# • /^ and JJL = h • v,
where g (resp. /?) is positive and locally integrable with respect to pt (resp v),
Hence (1 3.4.5) the function hg is locally ju-integrable, and we have \i = (kg) • /*,
which implies (13.15.3) that hg is equivalent (relative to /-i) to the function 1,
so that g(x) > 0 and h(x) = l/g(x) almost everywhere with respect to ju.
Conversely, suppose that v = g • JJL, with g(x) > 0 almost everywhere with
respect to p. Since (l/g(x))g(x) is equal to 1 almost everywhere with respect to
/z, the function l/g(x) is locally v-integrable, and we have \i = (l/#) • v
(13.14.5), so that the condition (a) is satisfied.

If \L and v satisfy the equivalent conditions of (13.15.6) they are said to be
equivalent positive measures on X. Clearly we have here an equivalence
relation on the set of positive measures on X. The notion of a measurable
function is the same for two equivalent measures (13.9.4).

(1 3.1 5.7) Ifjj, is any positive measure on X, there exists a continuous function h
such that h(x) > 0 for all x e X and such that the measure v = h • ju (which is
equivalent to \i by virtue of (13. 15. 6)) is bounded.

Let (Un) be an increasing sequence of relatively compact open sets in X
with the properties of (3.18.3), and for each n let/, be a continuous function on
X with values in [0, 1], such that/n(x) = 1 for all x e Un and /„(,*) = 0 for all
xeX — Un+1 (4.5.2). Let (an) be a sequence of real numbers >0 such that

00

£ an < +00. Then the series h = £ anfn is normally convergent in X (7.1),

n=l                                                n=l

hence h is continuous on X (7.2.1 ), and h(x) > 0 for all x e X. If v = h • n, then
v*(l) = J*/z dfjL£% anjfn diJi ((13.14.1) and (13.5.8)). If for example we take

if    \fndii>\9
\2'n       otherwise,
then we have ]T an < +00 and v*(l) < +00. This completes the proof.

(13.15.8) Let OO be any sequence of positive measures on X. Then there
exists a bounded positive measure v on X such that the relation v(N) = 0 is
equivalent to A£n(N) = 0 for all n (which implies (13.15.5) that each nn is a
measure with base v). Furthermore, if v' is another positive measure on X
having this property, then v and v' are equivalent.will be generalized later (13.15.8):